Evaluate : ` sin theta cos theta - (sin theta cos (90^(@) - theta) cos theta)/( sec (90^(@) - theta)) - (cos theta sin (90^(@) - theta) sin theta)/( cosec (90^(@) - theta))`

Options

A. `-1`
B. 2
C. 0
D. 1

Answer

To evaluate the expression

\[ \sin \theta \cos \theta - \frac{\sin \theta \cos(90^\circ - \theta) \cos \theta}{\sec(90^\circ - \theta)} - \frac{\cos \theta \sin(90^\circ - \theta) \sin \theta}{\csc(90^\circ - \theta)}, \]

we will follow these steps:

### Step 1: Simplify the first term
The first term is simply \(\sin \theta \cos \theta\).

### Step 2: Simplify the second term
We know that:
- \(\cos(90^\circ - \theta) = \sin \theta\)
- \(\sec(90^\circ - \theta) = \csc \theta\)

So, the second term becomes:
\[ -\frac{\sin \theta \sin \theta \cos \theta}{\csc \theta} = -\sin^2 \theta \cos \theta \cdot \sin \theta = -\sin^2 \theta \cos \theta. \]

### Step 3: Simplify the third term
We also know that:
- \(\sin(90^\circ - \theta) = \cos \theta\)
- \(\csc(90^\circ - \theta) = \sec \theta\)

Thus, the third term simplifies to:
\[ -\frac{\cos \theta \cos \theta \sin \theta}{\sec \theta} = -\cos^2 \theta \sin \theta \cdot \cos \theta = -\cos^2 \theta \sin \theta. \]

### Step 4: Combine all terms
Now we can combine all the terms:
\[ \sin \theta \cos \theta - \sin^2 \theta \cos \theta - \cos^2 \theta \sin \theta. \]

### Step 5: Factor out common terms
We can factor out \(\sin \theta \cos \theta\) from the expression:
\[ \sin \theta \cos \theta \left(1 - \sin \theta - \cos \theta\right). \]

### Step 6: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can rewrite \(1 - \sin^2 \theta - \cos^2 \theta\) as:
\[ \sin \theta \cos \theta \left(1 - 1\right) = \sin \theta \cos \theta \cdot 0 = 0. \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{0}. \]